A large variety of shaped tokamak discharges have been created in the Tokamak a Configuration Variable (TCV) with main parameters B-T<1.5 T, R-0=0.88 m, a<0.25 m, I(p)less than or equal to 1 MA [F. Hofmann, J. B. Lister, M. Anton et al., Plasma Phys. Controlled Fusion 36, B277 (1994)]. They include limited and diverted discharges with elongations up to 2.58 and triangularities between -0.7 and 1. Over the entire range of quasi-stationary sawtoothing Ohmic discharges investigated it is observed that the widths of plasma pressure profiles, [p]/p( 0) and the inversion radii scale as the widths of the current profiles, [j]/j(0), where [ ] refers to an average over the volume or, practically equivalently, the cross-sectional plasma area. These "profile consistency'' features are in apparent agreement with theoretical considerations and can be viewed as an extension of previous studies in circular discharges. The observations identify the parameter [j]/q(0)j(0) as the appropriate replacement for q(a) as a scaling parameter for arbitrary cross-sectional shapes. Using a simple current profile model with 0.8 less than or equal to q(0)less than or equal to 1 for the axial safety factor, it is shown that, for sawtoothing Ohmic discharges, such observations can be accounted for by the effects of sawtooth activity. (C) 1999 American Institute of Physics. [S1070-664X(99)04101-4].